In the attached report, the authors present the results of their research in the hyper-operations' field, so far, particularly concerning the infinite tower, the square super-root, as well as tetration and pentation. An attempt of introducing infinite or non-integer hyper-operation ranks is also included for future analysis.

The problem of the full complete definition, as well of the domain of convergence, of the "infinite tower" function is analyzed. Heuristic "piecewise" attempts are described to find the formula of the attractor curve near the origin, which defines an area "avoided" by all the tetration curves of the type: y(x) = x # n (for integer n, even and odd). A similar "attractor" curve was apparently parametrically described by Euler (documentation missing). The results so obtained is an experimental curve described by x = k.[y^(-y)-1], where k = 0.148398483.. . Some possible plots are shown, by using the Mathematica software package. Comparison is made of the behaviour of the square super-root together with some tetrational curves. Indication is shown of their domains and ranges of existence.

Some recursive hyper-operations' formulas are recalled and an analysis of the values assumed by the tetrational and pentational functions, for negative integer operands, is attempted. A new mathematical entity is defined, called Number Theta, as the result of an infinite iteration of the log operator, applied to numbers 0 or 1. A new constant "sigma", satisfying expression: sigma = sln sigma = e # sigma and also found as the asymptotic value of function e-penta-n, for n --> oo, is defined. A new limit hyper-operation, called omegation, together with its inverses (omegaroot and omegalog) is defined. The graph of this "functions" still need to be carefully analyzed. Attempts are also made for defining hyper-operations with non-integer ranks, based on some observations made by Prof. H. Paul Williams, LSE, among which a possible "sesquation" operation, with rank s=3/2, seems to be justified by the Gauss "arithmetic-geometric mean", that can be defined by using the "complete elliptical integral of the first kind". For more information on the subject see: http://mathworld.wolfram.com/Arithm...metricMean.html.

The authors are convinced that a lot of progress will be done in this area in the near future, with the successful achievements obtained in the framework of "Fractional Calculus" and of the analytic continuation of operators (see: http://mathworld.wolfram.com/FractionalCalculus.html). They also believe that hyper-operations will be an excellent tool for NKS developments and a very useful framework, in Science and Technology, particularly for designing new data formats for storing and handling extremely large numbers and practically avoiding computing overflow. It goes without saying that almost all hyper-operations commands can easily be included in Mathematica.

A Poster will jointly be presented by the authors at the International Conference of Mathematicians (ICM-06), Madrid, 20-30 August, 2006 (Hyper-operations as a Tool for Science and Engineering).